So here's how I would think about it intuitively:

We can create a balanced partitioning of the 300 turkeys with a 300 bit random number having an equal number of 1's and 0's.

Now suppose I randomly pick 300 bit number, still with equal 0's and 1's, but this time the first 20 bits are always 0's and the last 20 bits are always 1's. In this scenario, only the middle 260 bits (turkeys) are randomly assigned, and the remaining 40 are deterministic.

We can quibble over what constitutes an "enormous" bias, but the scenario above feels like an inadequate experiment design to me.

As it happens, log2(260 choose 130) ~= 256.

> Are there any non-cryptographic examples in which a well-designed PRNG with 256 bits of well-seeded random state produces results different enough from a TRNG to be visible to a user?

One example that comes to mind is shuffling a deck of playing cards. You need approximately 225 bits of entropy to ensure that every possible 52 card ordering can be represented. Suppose you wanted to simulate a game of blackjack with more than one deck or some other card game with more than 58 cards. 256 bits is not enough there.